Because the derivative of a function

*y* =

*f*(

*x*) is itself a function

*y′* =

*f′*(

*x*), you can take the derivative of

*f′*(

*x*), which is generally referred to as the

*second derivative of f(x)* and written

*f“*(

*x*) or

*f* ^{2}(

*x*). This differentiation process can be continued to find the third, fourth, and successive derivatives of

*f*(

*x*), which are called

**higher order derivatives** of

*f*(

*x*). Because the “prime” notation for derivatives would eventually become somewhat messy, it is preferable to use the numerical notation

*f*(

^{n })(

*x*) =

*y*(

^{n }) to denote the

*n*th derivative of

*f*(

*x*).

**Example 1:** Find the first, second, and third derivatives of *f*( *x*) = 5 *x* ^{4} − 3x ^{3} + 7x ^{2} − 9x + 2.

**Example 2:** Find the first, second, and third derivatives of *y* = sin ^{2} *x*.

**Example 3:** Find *f* ^{(3)} (4) if .